Question

If the angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is ______.

Options

•  200 m

•  500 m

•  30 m

• 400 m

Answer 1

If the angle of elevation of a cloud from a point 200 m above a lake is 30° and the angle of depression of its reflection in the lake is 60°, then the height of the cloud above the lake is 400 m.

Explanation:

Let the height of the point of observation above the lake be H = 200 m.

Let the height of the cloud above the lake be h.

Let the horizontal distance between the observer and the cloud be d.

Angle of Elevation (30°):

The height of the cloud above the point of observation is (h – 200) m.

Using the definition of tangent:

`tan(30^\circ) = "Opposite"/"Adjacent" = (h - 200)/d`

`= 1/sqrt3 = (h - 200)/d`

d = (h – 200) √3     ...(Eqn. 1)

Angle of Depression (60°):

The reflection of the cloud in the lake is at a depth equal to its height above the water, which is h meters below the surface.

The observer is 200 m above the lake, so the total vertical distance from the observer to the reflection is (200 + h) m.

`tan(60^\circ) = "Opposite"/"Adjacent" = (h - 200)/d`

`sqrt3 = (200+h)/d`

`d = (200+h)/sqrt3`     ...(eqn. 2)

Solving for h:

Equating the two expressions for d:

`(h - 200) sqrt3 = (200+h)/sqrt3`

Multiply both sides by √3

3 ( h - 200) = 200 + h

3h - 600 = 200 + h

3h - h = 200 + 600

2h = 800

h = 400